88.8.5 problem 5
Internal
problem
ID
[24013]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Exercise
at
page
44
Problem
number
:
5
Date
solved
:
Thursday, October 02, 2025 at 09:54:06 PM
CAS
classification
:
[_rational]
\begin{align*} y+6 x y^{3}-4 y^{4}-\left (2 x +4 x y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 694
ode:=y(x)+6*x*y(x)^3-4*y(x)^4-(2*x+4*x*y(x)^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{1}/{3}}+\frac {\left (3 x^{2}+c_1 \right )^{2}}{\left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{1}/{3}}}+3 x^{2}+c_1}{12 x} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{2}/{3}}}{24}+\frac {3 \left (\frac {2 \left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{1}/{3}}}{3}+\left (i \sqrt {3}-1\right ) \left (x^{2}+\frac {c_1}{3}\right )\right ) \left (x^{2}+\frac {c_1}{3}\right )}{8}}{x \left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{2}/{3}}}{24}+\frac {3 \left (\frac {2 \left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{1}/{3}}}{3}+\left (-i \sqrt {3}-1\right ) \left (x^{2}+\frac {c_1}{3}\right )\right ) \left (x^{2}+\frac {c_1}{3}\right )}{8}}{x \left (27 x^{6}+27 c_1 \,x^{4}+12 \sqrt {3}\, \sqrt {\frac {27 x^{6}+27 c_1 \,x^{4}+9 c_1^{2} x^{2}+c_1^{3}+108 x^{3}}{x}}\, x^{2}+9 c_1^{2} x^{2}+c_1^{3}+216 x^{3}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 15.302 (sec). Leaf size: 638
ode=( y[x]+6*x*y[x]^3-4*y[x]^4 )-( 2*x+4*x*y[x]^3 )*D[y[x],{x,1}]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {3 x^2+\sqrt [3]{27 x^6+27 c_1 x^4+216 x^3+9 c_1{}^2 x^2+12 \sqrt {3} \sqrt {x^3 \left (27 x^6+27 c_1 x^4+108 x^3+9 c_1{}^2 x^2+c_1{}^3\right )}+c_1{}^3}+\frac {\left (3 x^2+c_1\right ){}^2}{\sqrt [3]{27 x^6+27 c_1 x^4+216 x^3+9 c_1{}^2 x^2+12 \sqrt {3} \sqrt {x^3 \left (27 x^6+27 c_1 x^4+108 x^3+9 c_1{}^2 x^2+c_1{}^3\right )}+c_1{}^3}}+c_1}{12 x}\\ y(x)&\to \frac {2 \left (3 x^2+c_1\right )-\frac {i \left (\sqrt {3}-i\right ) \left (3 x^2+c_1\right ){}^2}{\sqrt [3]{27 x^6+27 c_1 x^4+216 x^3+9 c_1{}^2 x^2+12 \sqrt {3} \sqrt {x^3 \left (27 x^6+27 c_1 x^4+108 x^3+9 c_1{}^2 x^2+c_1{}^3\right )}+c_1{}^3}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{27 x^6+27 c_1 x^4+216 x^3+9 c_1{}^2 x^2+12 \sqrt {3} \sqrt {x^3 \left (27 x^6+27 c_1 x^4+108 x^3+9 c_1{}^2 x^2+c_1{}^3\right )}+c_1{}^3}}{24 x}\\ y(x)&\to \frac {2 \left (3 x^2+c_1\right )+\frac {i \left (\sqrt {3}+i\right ) \left (3 x^2+c_1\right ){}^2}{\sqrt [3]{27 x^6+27 c_1 x^4+216 x^3+9 c_1{}^2 x^2+12 \sqrt {3} \sqrt {x^3 \left (27 x^6+27 c_1 x^4+108 x^3+9 c_1{}^2 x^2+c_1{}^3\right )}+c_1{}^3}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^6+27 c_1 x^4+216 x^3+9 c_1{}^2 x^2+12 \sqrt {3} \sqrt {x^3 \left (27 x^6+27 c_1 x^4+108 x^3+9 c_1{}^2 x^2+c_1{}^3\right )}+c_1{}^3}}{24 x}\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(6*x*y(x)**3 - (4*x*y(x)**3 + 2*x)*Derivative(y(x), x) - 4*y(x)**4 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out